An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Let y = tanh − 1 ( x ) , then x = tanh ( y ) .
Express tanh ( y ) in terms of exponentials: x = e 2 y + 1 e 2 y − 1 .
Solve for e 2 y : e 2 y = 1 − x 1 + x .
Solve for y : y = 2 1 ln ( 1 − x 1 + x ) . Therefore, tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x ) .
Explanation
Problem Setup We are given the equation tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x ) for − 1 < x < 1 , and we want to prove it using the method outlined in the problem. The method starts by letting y = tanh − 1 ( x ) .
Expressing x in terms of exponentials Then, we have x = tanh ( y ) . We know that tanh ( y ) = c o s h ( y ) s i n h ( y ) . Also, we know that sinh ( y ) = 2 e y − e − y and cosh ( y ) = 2 e y + e − y . Therefore, tanh ( y ) = 2 e y + e − y 2 e y − e − y = e y + e − y e y − e − y . Multiplying the numerator and denominator by e y , we get tanh ( y ) = e 2 y + 1 e 2 y − 1 . So, x = e 2 y + 1 e 2 y − 1 .
Solving for y Now, we solve for y in terms of x . We have x = e 2 y + 1 e 2 y − 1 . Multiplying both sides by e 2 y + 1 , we get x ( e 2 y + 1 ) = e 2 y − 1 . Expanding, we have x e 2 y + x = e 2 y − 1 .
Isolating the exponential term Rearranging the terms, we get e 2 y − x e 2 y = x + 1 , which can be written as e 2 y ( 1 − x ) = 1 + x .
Finding e^(2y) Dividing both sides by 1 − x , we get e 2 y = 1 − x 1 + x .
Taking the logarithm Taking the natural logarithm of both sides, we have 2 y = ln ( 1 − x 1 + x ) .
Final Result Finally, dividing by 2, we get y = 2 1 ln ( 1 − x 1 + x ) . Since we initially let y = tanh − 1 ( x ) , we have tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x ) . This completes the proof.
Examples
The hyperbolic tangent inverse function is useful in various fields such as physics and engineering. For example, it can be used to calculate the rapidity of a particle in special relativity, which is a measure of its velocity relative to the speed of light. Also, in machine learning, the hyperbolic tangent function and its inverse are used in neural networks as activation functions to introduce non-linearity into the model. Understanding these functions helps in modeling complex relationships in data.
To find out how many electrons flow through a device delivering a current of 15.0 A for 30 seconds, we calculate the total charge using the formula Q = I × t , which gives us 450 C. Then, we divide this charge by the charge of a single electron (approximately 1.6 × 1 0 − 19 C ), leading to approximately 2.81 × 1 0 21 electrons flowing through the device. Thus, around 2.81 × 1 0 21 electrons pass through the device during this time period.
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